Scrutinizing joint remote state preparation under decoherence

This research examines the effect of an open system containing the squeezed generalized amplitude damping channel on the joint remote preparation quantum communication protocol using a maximally entangled two-qubit state. Our findings indicate that the fidelity of a quantum system in contact with a non-zero temperature thermal bath can be enhanced by varying the squeezing parameters. These parameters include the squeezing phase of the channel \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi$$\end{document}Φ and the amount of squeezing of the channel r.

www.nature.com/scientificreports/ to both amplitude damping and squeezing noise. In this research, we consider the effect of the SGAD channel on the fidelity of information transmitted in the JRSP protocol of an arbitrary two-qubit state. The JRSP protocol has been extensively explained by other authors; our aim in this work is mainly to determine the effect of SGAD on the already established JRSP model of an arbitrary 2-qubit state. For quantum systems that have interactions with their environment, the origin of the noise is the decoherence effect, which is brought about by the system's interaction with the channel. The advancement over time of a quantum system that is continuously influenced by its environment can be defined using the master's equation in the Lindblad form 45 for the density operator ρ(t) 3,46,47 within the framework of Markov and Born approximations. Also in JRSP architecture, the senders and receivers are required to share a quantum state with one another as a communication channel to finalize the preparation. In a practical JRSP, the channel state must be generated by one of the parties that make up the JRSP, and the resulting qubits must be shared with the respective members through a noisy channel. This procedure changes the pure channel state into a mixed state.
In this research, Kraus operators are employed to give a representation of the consequences of the noisy channel. One major type of noise is considered; the squeezed generalized amplitude damping noise, which is a generalized type of noise that may be used to model the amplitude damping (AD) and the generalized amplitude damping (GAD) noises by assigning particular values to some of their parameters. Fidelity was used to compare the closeness of the final state received by the receiver to the initial state sent by the sender and was also used to quantify how much information was lost in the JRSP process. Finally, the results are discussed and comparisons are made between the AD, GAD, and SGAD channels, and we are able to show how each of the parameters of the SGAD influenced the fidelity of the JRSP system. This paper is arranged as follows: In Section "Review of joint remote preparation of two qubit system", we explore the JRSP of a two-qubit system. In Section "Fidelity computation procedure", we introduce the fidelity computation framework. In Section "Kraus operators, fidelity computation of SGAD channel and associated dissipative channels", we compute the fidelities of the SGAD channel and other associated dissipative channels, and in Section "Discussion and conclusion", we discuss our results and conclude.

Review of joint remote preparation of two qubit system
In this work, we use the Joint Remote State Preparation as introduced by Wang et al. 48 . They used a six-qubit cluster state to prepare an arbitrary two-qubit state. Their scheme is as follows: If we consider two participants named Alice and Bob who are interested in assisting a third remote participant named Caleb, they prepare an arbitrary two-qubit quantum state, which may be described as (1) where coefficients a i (i = 0, 1, 2, 3) in (1) are real and satisfy the normalization condition a 2 0 + a 2 1 + a 2 2 + a 2 3 = 1 , as well as θ j ∈ [0, 2π](j = 0, 1, 2, 3) . The information in |B� is shared between Alice and Bob; Alice holds the amplitude information a i and Bob holds on to the phase information θ j . To be able to send the original state to Caleb, they require a quantum state as a communication channel, such as (2) where Alice's particles are (1,4), Bob's (2,5), and Caleb's (3,6). Alice and Bob choose an orthogonal basis to measure their local qubits. The forms of the measurement bases are as follows: we have |ρ 1 � 14 ,|ρ 2 � 14 ,|ρ 3 � 14 , |ρ 4 � 14 .
As shown in Table 1, Caleb can easily obtain the original message by performing simple unitary operations on the state he receives for 4 out of the 16 possible outcomes. This shows that Caleb is able to obtain the original message only 25% of the time, assuming the system is a closed system with no interaction with its environment. However, for an open system that interacts with its environment, it is susceptible to noise, which may affect the entanglement of the qubits under consideration and thus affect the fidelity of the information that reaches Caleb. In the research, we are investigating the fidelity of the information that reaches Caleb, assuming information from Alice and Bob gets to Caleb through a noisy channel that is characterized by SGAD noise.

Fidelity computation procedure
The final density matrix at Caleb's side is represented by Eq. (6) as shown in Eq. (6), Tr 14,25 , represents a partial trace over Bob's and Alice's particles, which are particles (1,4) and (2,5) for Bob and Alice, respectively. The unitary operator U 0 accounts for the sequence of events in the JRSP communication protocol. It is worthy of note that in a perfect JRSP system that is not influenced by noise, the density matrix expressed in Eq. (6) will be very similar to that of the initial state that was transmitted; however, due to the influence of the noisy channel, there is always a notable difference. To investigate the effect of noise on the JRSP protocol, we only consider situations where the JRSP has reached the desired end, as shown in Table 1. In Eq. (6), the value of U 0 is then expressed as shown in Eq. (7).
where kǫ(1, 2, 3, 4) and σ 1 36 U 0 represent operations that can be carried out by Caleb on his received quantum state. To determine the closeness of the state received by Caleb to the original state shared between Alice and Bob and transmitted, fidelity is used. Since the original state transmitted is given by the Eq. (1), the fidelity of the JRSP system may then be expressed as given in Eq. (8): When F = 1, it implies a perfect JRSP communication system where the received information is exactly the way it was transmitted, while a value of F less than 1 implies that some transmitted information has been lost in transit. The lower the value of the fidelity, the more information that has been lost.

Kraus operators, fidelity computation of SGAD channel and associated dissipative channels
The SGAD channel is a dissipative channel that is a generalization of the Amplitude Damping (AD) and Generalized Amplitude Damping (GAD) channels with a squeezing effect. The squeezing effect, which is a quantum asset, delivers added advantages over GAD channels. Therefore, investigating the SGAD channel enables obtaining results about channels involving both non-zero temperatures as well as squeezing parameters 49 . A squeezed reservoir can be devised based on the framework of the installation of a squeezed light field 50 . Experiments investigating the squeezed light atom have been embarked on by Refs. 51,52 . A number of authors agree that a benefit of a squeezed thermal bath is that the rate of degeneration of quantum coherence is decreased, which implies a conservation of quantum resources [53][54][55][56] . SGAD has also been proven to alter the progression of the geometric phase of a two-level atomic system 57 . It has also been observed that the SGAD channel has restorative attributes 57,58 .
A number of experiments have been carried out on the SGAD noise model. For example SGAD has been modelled using squeezed light field 50,59 , subthreshold Optical Parametric Oscillator (OPO) 51,52 , beam splitters 60 and laser cooled trapped ions 61 .
To obtain the fidelity for the SGAD channel, first the Kraus operator for the SGAD channel acts on the qubits (1, 4) and (2,5) corresponding to Alice's and Bob's particles, as shown below: By substituting the results of Eq. (13) into Eq. (6) and evaluating using Eqs. (7) and (8) we obtain the equation representing the fidelity of the SGAD channel. The equation, however, is quite complex and involves many variables that, due to space constraints, cannot be expressed in this paper. However, the graph expressing the attributes of the SGAD channel is shown in Fig. 1.

Amplitude damping (AD)
Channel. This is one of the channels that can be modeled using the SGAD by assigning some specific values to some of the parameters of the SGAD. The AD channel acts like the interaction of a quantum system with a vacuum bath 49 , It presents the rate of energy loss in a quantum state due to its interaction with a vacuum bath. A lot of work has been carried out on this noise model, and a lot of applications have been found for the model [64][65][66][67][68][69] . For example, it is used in the basic conceptual structure of the weak Born-Markov approximation in analyzing the spontaneous emission of a photon by a two level system into a photon environment at low temperature. The Kraus operators for the AD channel can be written as 70 : where the decoherence rate A : (0 ≤ ≤ 1) , represents the likelihood of error when particles traverse an AD noisy channel. As declared earlier, AD noise has an effect on only Alice's and Bob's qubits, which are (1, 4, 2, 5) Qubits 3 and 6 are not affected by the noisy channel. The effect of the AD channel on the quantum state of the information being transmitted can be represented by the following expression: , the noise parameter Q is set to 1, the parameter µ is set to 0, the parameter v is such that v = and the squeezing parameters r and are set to 0. Making these substitutions in Eq. (9) will reduce it to Eq. (14). We obtain the fidelity of the system, which is a measure of how close the final state is to the initial state as: According to Eq. (16), the fidelity of the AD channel is solely determined by the amplitude damping noise parameter ( A ). From Fig. 2, it can be seen that when A has a maximum value, which is when A = 1 , we have   Figure 1. This plot shows the attributes of the SGAD channel. The plots show the variation of fidelity of the SGAD channel at different temperatures and different values of the squeezing parameters r and as indicated on the plots. When the squeezing parameters r and are changed while the other parameters remain constant, the fidelity swings high and low around the same value limits. Looking at plots (b), (c), and (e), it is clear that when r = 1 and = 90 , 180, and 360, respectively, an increase in fidelity with an increase in temperature is possible. Also, looking at plots (a) and (d), it is observed that when r = 0.5 and = 0 and 270, fidelity is sustained for higher temperatures at which it normally would have been completely diminished. This demonstrates that squeezing parameters can be used to improve fidelity in non-zero temperature thermal baths.
To arrive at the plots, the following values were assigned to other SGAD parameters: ω = 0.5 , Q=0.5, γ = 0.5 . where µ , v, and are functions of temperature T, γ , ω and squeezing parameters r as expressed in Eqs. (10), (11) and (12) and units are such that ≡ k ≡ 1. www.nature.com/scientificreports/ the fidelity F AD = 1 8 which is the minimum fidelity that can be attained in the AD channel, while when A = 0 the fidelity is equal to 1, which signifies a perfect JRSP.
Generalized amplitude damping channel. This is another channel that may be modeled using the SGAD by assigning specific values to some of its parameters. In a GAD channel, the quantum system loses and gains excitation by interacting with the environment. The GAD channel is utilized in cloning the spontaneous emission of a particle subjected to a vacuum bath with a temperature greater than zero. Equation (17) gives the Kraus operators for the GAD channel 70,71 .
To obtain the GAD channel Kraus operators from Eq. (17), we substitute = 0 , µ = 0 , and v = . The effect of the GAD channel on the quantum state of the information being transmitted can be represented by the expression given below: By substituting the result from the computation of Eq. (18) into Eq. (6) and evaluating Eqs. (7) and (8) Figure 2. The fidelity of Amplitude Damping model shows decreasing fidelity with an increase in the noise parameter A . This means that the higher the value of the noise parameter A the greater the loss of information along the channel, leading to a higher difference between the information sent by Alice and Bob and the information received by Caleb. The plot clearly shows that the channel's fidelity is highest, which is 1, when A is 0 and lowest, which is ( 1 8 ), when A is 1.
, which always returns 1 8 regardless of Q. When both Q and are equal to 1, the fidelity of the channel is F G = 1 8 which is it's minimum fidelity. G on the other hand, is temperature-dependent and can be calculated as . With these changes, using the Eq. (19) and γ = 0.05 , The expression for the fidelity is obtained, which gives a very complex and long equation that may not be appropriate for this article, but the properties of the fidelity are as shown in Fig. 3. By making ω constant and varying the temperature and the γ parameter, we observe, as shown in Fig. 4) that the fidelity of the given channel decreases with increasing γ .
Taking the values of ω and Q to be equal to 1, we obtain the equation for the fidelity of the channel to be given by Eq. (20).

Discussion and conclusion
The fidelities of joint remote preparation of a maximally entangled two qubit state |B� = a 0 e iθ 0 |00� + a 1 e iθ 1 |01� + a 2 e iθ 2 |10� + a 3 e iθ 3 |11� where a 0 = a 1 = a 2 = a 3 = 1 2 in contact with an SGAD noisy channel, and other dissipative channels, namely the amplitude damping (AD) and the generalized amplitude damping (GAD), have been examined. We found that as the noise parameter A increases, the fidelity of the AD channel decreased, with a minimum value of 1 8 when A = 1 and a maximum value of 1 (which denotes a perfect JRSP) when A = 0 . For GAD with a noise parameter G where G is dependent on temperature T, frequency of photons ω and spontaneous emission rate γ , it was observed that fidelity of the channel decreased with increase in temperature, ω and γ still having a value of 1 8 when G = 1 and having the attributes of a perfect JRSP when G = 0 . For the SGAD channel, though it is already established that the fidelity of a JRSP quantum protocol in contact with a thermal bath decreases with increase in temperature, it was observed that it is possible to get an opposing result for particular values of the squeezing parameters r and . When r = 1 and Phi = 90 , 180, and 360, an increase in fidelity with increasing temperature is observed, while when r = 0.5 and = 0 and 270, it is observed that fidelity is sustained at higher temperatures at which it would normally have been diminished.  It can be seen that the fidelity of the JRSP system decreases with increasing values of the probability parameter Q until a value of 0.5, at which point it starts to increase again. The fidelity is maximum at Q = 0 and Q = 1 and minimum at Q = 0.5 when G = 0 . At G = 1 , the fidelity of the system is 1 8 regardless of the value of Q, which is the minimum fidelity that can be obtained from the JRSP system. This is very similar to an AD channel. However, it is worthy of note that in a GAD channel, the noise parameter G is dependent on temperature. To arrive at these plots, the values of the squeezing parameters r and were set to 0, µ was set to 0, and v was set to be equal to which is equal to 1 exp ω T −1 while Q was set to 0.5. The plots a-f shows how fidelity changing with temperature and frequency at spontaneous emission rates γ of 0.5, 1.0, 1.5, 2.0, 2.5 and 3.0, respectively in units such that ≡ k ≡ 1 . The plots show that increases in temperature and ω lead to decreases in fidelity; however, the rate of decrease in fidelity with an increase in temperature is higher than the rate of decrease in fidelity with an increase in ω . The plots also show that, generally, the fidelity decreases with an increase in the spontaneous emission rate γ.